nLab
module algebra

A left module kk-algebra is a kk-algebra AA equipped with a left Hopf action (also sometimes said module algebra action) ▹:B⊗A→A\triangleright: B\otimes A\to A of a kk-bialgebra BB. Sometimes one talks about module algebras meaning a monoid with Hopf actions of a bimonoid in a more general symmetric monoidal category; a module monoid would be a better term if the category is not kk-linear.